Add Brownian motion based primitive kernels

[SamuelBrand1]

Hi there,

I think one gap in AutoGP is that it doesn't offer non-stationary kernel primitives, e.g. Brownian motion or integrated brownian motion (IBM).

In particular, cubic spline regression for 1d data (i.e. time series) can be recovered by the kernel compostion: CONST + LINEAR + IBM. I think thats quite powerful to expose as something that can be learnt using involutive MCMC moves on the structure.

I do note however, that adding a new kernel primitive would require replumbing the GPConfig struct, so its a bit of a lift.

[SamuelBrand1]

Looking at this... it looks very hard to achieve with the current SMC proposal system which seems to hard-code the precise primitive/leaf kernels.

@fsaad would you be interested in a PR on this, aimed at making the SMC proposals more extensible along with adding a couple more primitive kernels?

[fsaad]

I think one gap in AutoGP is that it doesn't offer non-stationary kernel primitives

The LIN kernel, and compositions thereof, define nonstationary Gaussian processes, see here.

integrated brownian motion (IBM)

By IBM, do you mean Brownian Integral Kernel? If so, then it appears incompatible with AutoGP. Our covariance kernels take the form $$k(t,t')$$ for two real time points $t,t'$, whereas the BIK is defined over time intervals $k((s,e), (s',e'))$ where $(s,e) \in \mathbb{R}^2$ is a time a interval for the start and end time.